Cointegration Vector Estimation by Panel DOLS and Long-run...

Cointegration Vector Estimation by Panel

DOLS and Long-run Money Demand*

Nelson C. Mark�,�,§ and Donggyu Sul–

�Ohio State University (e-mail: [email protected])

�University of Notre Dame§NBER

–University of Auckland (e-mail: [email protected])

Abstract

We study the panel dynamic ordinary least square (DOLS) estimator of a

homogeneous cointegration vector for a balanced panel of N individuals

observed over T time periods. Allowable heterogeneity across individuals

include individual-specific time trends, individual-specific fixed effects and

time-specific effects. The estimator is fully parametric, computationally

convenient, and more precise than the single equation estimator. For fixed N

as T fi 1, the estimator converges to a function of Brownian motions and theWald statistic for testing a set of s linear constraints has a limiting v2(s)distribution. The estimator also has a Gaussian sequential limit distribution

that is obtained first by letting T fi 1 and then letting N fi 1. In a series ofMonte-Carlo experiments, we find that the asymptotic distribution theory

provides a reasonably close approximation to the exact finite sample

distribution. We use panel DOLS to estimate coefficients of the long-run

money demand function from a panel of 19 countries with annual

observations that span from 1957 to 1996. The estimated income elasticity

is 1.08 (asymptotic s.e. ¼ 0.26) and the estimated interest rate semi-elasticity

is )0.02 (asymptotic s.e. ¼ 0.01).

*This paper was previously circulated under the title ‘A Computationally Simple CointegrationVector Estimator for Panel Data’. For valuable comments on earlier drafts, we thank Ronald Bewley,Roger Moon, Peter Phillips, seminar participants at Georgetown University, Ohio State University,the 2001 New Zealand Econometric study group meeting, the University of California at SantaBarbara, the University of Southern California, and an anonymous referee. The usual disclaimerapplies.

OXFORD BULLETIN OF ECONOMICS AND STATISTICS, 65, 5 (2003) 0305-9049

655� Blackwell Publishing Ltd, 2003. Published by Blackwell Publishing Ltd, 9600 Garsington Road, Oxford OX4 2DQ, UK

and 350 Main Street, Malden, MA 02148, USA.

I. Introduction

This paper considers the extension of the single equation dynamic ordinary

least squares (DOLS) method of Saikkonen (1991) and Stock and Watson

(1993) for estimating and testing hypotheses about a cointegrating vector to

panel data. We call the estimator panel DOLS. We discuss its limit distribution

and apply it to estimate the long-run money demand function using a panel

data set of 19 countries with annual observations spanning from 1957 to 1996.

Panel DOLS is fully parametric and offers a computationally convenient

alternative to the panel ‘fully modified’ OLS estimator proposed by Pedroni

(1997) and Phillips and Moon (1999). Properties of panel DOLS, when there

are fixed effects in the cointegrating regression, have been discussed by Kao

and Chiang (2000). We take this to be the starting point for our analysis. In our

environment, the cointegrating vector is homogeneous across individuals but

we allow for individual heterogeneity through disparate short-run dynamics,

individual-specific fixed effects and individual-specific time trends. Moreover,

we permit a limited degree of cross-sectional dependence (CSD) through the

presence of time-specific effects.

We present two limit distributions for panel DOLS. The first limit

distribution is obtained for a fixed number of cross-sectional units N, letting

T fi 1. In this case, panel DOLS converges in distribution to a function ofBrownian motions and the Wald statistic for testing a set of s linear constraints

has a limiting v2(s) distribution. This limit theory seems well suited for manyapplied macroeconomic or international problems. Here, researchers often

have available panel data sets of moderate N but much larger T. With the

passage of time, these data sets will gain time-series observations but they are

unlikely to acquire many more cross-sectional units.1 We also obtain the

sequential limit distribution by first letting T fi 1 for fixed N, and then

letting N fi 1 as proposed by Phillips and Moon (1999). Here, panel DOLS

has a limiting Gaussian distribution and as in the fixed N case, the Wald

statistic has a limiting chi-square distribution. In the absence of linear trends in

the cointegrating regression, the sequential limiting normality of the estimator

is theoretically interesting but has less practical import because the limit

distribution of the test statistics is identical to the T fi 1 distribution with

fixed N. However, when linear trends are present, the sequential limit theory

produces considerable simplifications. Here, the estimator of the cointegration

1For example, if the observational unit is a national economy, the total number of countries mayfluctuate over time, but is unlikely to go to infinity. While the break-up of the Soviet Union createdseveral new economies, the opposite trend is at work in Europe where the EMU may eventuallycombine to form a single economic unit. But beyond this, researchers typically choose to groupcountries into classes that share common characteristics such as income levels, stages of developmentor geography which often result in panels with 5–20 individuals.

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� Blackwell Publishing Ltd 2003

vector and the time-trend slope coefficients remain correlated for fixed N as

T fi 1 but are asymptotically uncorrelated when T fi 1 then N fi 1.As single equation cointegration vector estimators are super consistent, it is

natural to ask what is to be gained by using the panel estimator. The answer is

that super consistency means that convergence towards the asymptotic

distribution occurs at rate T but it says nothing about the sampling variability

of the estimator for a fixed value of T. In fact, the statistical properties of

single-equation cointegration-vector estimators can be quite poor when

applied to sample sizes associated with macroeconomic time series typically

available to researchers (e.g. Inder, 1993; Stock and Watson, 1993). Moreover,

even limited amounts of heterogeneity in the short-run dynamics across

individuals can generate considerable disparities in single-equation DOLS

estimates of the true homogeneous cointegration vector. In these situations,

combining cross-sectional and time-series information in the form of a panel

can provide much more precise point estimates of the cointegration vector

with reasonably accurate asymptotic approximations to the exact sampling

distribution. In a series of Monte-Carlo experiments, we study the small

sample performance of panel DOLS and compare it with single-equation

DOLS. Panel DOLS generally performs well under the short-run dynamic

designs that we consider and can attain a striking improvement in estimation

precision over that of single-equation DOLS with even a modest number of

cross-sectional units.

We then apply panel DOLS to estimate the long-run demand for M1

money. The countries in our study are Austria, Australia, Belgium, Canada,

Denmark, France, Finland, Germany, Iceland, Ireland, Japan, Norway, New

Zealand, the Netherlands, Portugal, Spain, Switzerland, the United Kingdom,

and the United States. Here, we build on the time-series contributions by

Baba, Hendry and Starr (1992), Ball (1998), Hoffman, Rasche, and Tieslau

(1995), Lucas (1988) and Stock and Watson (1993), and the cross-sectional

studies by Mulligan and Sala-i-Martin (1992), and Mulligan (1997), most of

which has focused on US data.2

The studies cited above report conflicting results along three dimensions.

First, point estimates from time-series studies exhibit substantial dependence on

the sample period of the data. Income elasticity estimates from post-WWII US

data typically lie well below 1 – which implies the existence of economies of

scale in money management – whereas estimates obtained from pre-WWII

observations or those that combine pre- and postwar observations are generally

close to 1. Using annual US data spanning from 1903 to 1987, Stock and

Watson’s (1993) DOLS estimate of the income elasticity is 0.97. When the

sample spans from 1903 to 1945, their estimate is 0.89 but drops to 0.27 when

2Less recent cross-sectional studies include Meltzer (1963) and Gandolfi and Lothian (1976).

657Cointegration vector estimation by panel DOLS and long-run money demand


the data span from 1946 to 1987. Ball (2001) extends these data and obtains an

estimate of 0.42 when the observations span from 1946 to 1996. Using annual

observations from 1900 to 1958, Lucas’s (1988) estimate of theM1 (permanent)

income elasticity is 1.06 and his estimate of the (short-term) interest rate semi-

elasticity is)0.07. Using data spanning from 1958 to 1985, his income elasticityestimate drops to 0.21 and his interest semi-elasticity estimate is )0.01.Secondly, there is tension generated by the large difference between the

estimates from time-series studies and those from postwar cross-section studies.

Mulligan and Sala-i-Martin’s (1992) estimates from a 1989 cross-sectional data

set from the Survey of Consumer Finances range between 0.82 and 1.37.

Mulligan (1997) runs OLS on for a panel of 12,000 firms observed from 1961 to

1992 and obtains an income-elasticity estimate of 0.83. Thirdly, there is

substantial cross-country variation even amongst economies of similar income

levels and financial market development. In our analysis, single-equationDOLS

with trend gives such disparate income elasticity estimates as )1.23 for NewZealand and 2.42 for Canada. The corresponding interest rate semi-elasticity

estimates range from 0.02 for Ireland (which has thewrong sign) to)0.09 for theUK. When trends are omitted, the income elasticity estimates range from 0.13

for Belgium to 2.64 for Norway and the interest semi-elasticity estimates range

from 0.02 for Ireland to )0.16 for Norway.With only 40 annual observations, the heterogeneity that we observe in the

point estimates may plausibly have been generated from an underlying data

generating process with a homogeneous cointegration vector and heterogeneous

short-run dynamics. When we include heterogeneous linear trends and estimate

the cointegrating vector by panel DOLS, we obtain a point estimate of the

income elasticity of 1.08 (asymptotic s.e. ¼ 0.26) and a point estimate of the

interest semi-elasticity of )0.02 (asymptotic s.e. ¼ 0.01). Moreover, these

estimates, which are more in line with those from cross-sectional studies on US

data, are stable as the span of the time-series dimension is varied and are

reasonably robust to the inclusion or omission of heterogeneous linear trends.

The remainder of the paper is organized as follows. The next section des-

cribes the representation of the non-stationary panel data and regularity con-

ditions assumed in the paper. Section III describes the panel DOLS estimator

and discusses its asymptotic properties. Section IV reports the results of a

Monte-Carlo experiment to examine the small sample performance of the panel

DOLSestimator and the accuracyof the asymptotic approximations. In sectionV

we present our long-run money demand study and section VI concludes the

paper. Proofs of propositions and supplementary results from themoney demand

study are given in an appendixwhich is available upon request from the authors.3

3Alternatively, the appendix can be downloaded from the http://www.wcon.ohio-state.edu/Mark/nmark.htm.

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II. Representation of cointegrated observations in panel data

Consider a balanced panel of individuals indexed by i ¼ 1,…,N observed

over time periods t ¼ 1,…, T. Vectors are underlined and matrices appear inbold face. W(r) is a vector standard Brownian motion for 0 £ r £ 1, and [Tr]denotes the largest integer value of Tr for 0 £ r £ 1. As is standard in theliterature, we will drop the notational dependence on r and write

R 10W ðrÞdr asR

W andR 10W ðrÞdW ðrÞ0 as

RW dW 0. Scaled vector Brownian motions are

denoted by B ¼ KW where K is a scaling matrix. For any matrix A, ||A||

denotes the Euclidian norm, [Tr(A0A)]1/2.We will be working with double indexed sums. In some instances – to

deal with individual-specific fixed effects or common-time effects – these

sums will involve transformations of the original observations. To handle

such situations, we generically denote the sample cross-moment matrix

averaged over N and T2 as MNT and let the precise definition depend upon

the particular model under consideration. Also, we generically denote the

limit of the moment matrix as T fi 1 for any given N by MN. As N fi 1,MN need not converge to a constant and we denote this limit as MN .

Similarly, our generic notation for the sample cross-product vector between

the regressors and the equilibrium error is mNT, and the limit for fixed N as

T fi 1 is mN.

As the model we study allows for individual specific effects, perhaps

it would be more accurate to call the estimator dynamic least squares dummy

variable (LSDV). However, in the interests of simplicity, we will refer to the

estimator as panel DOLS.

(i) Triangular representation

Let fðyit; x0itÞ0g be a (k + 1) dimensional vector of observations where yit is a

scalar and xit is a k-dimensional vector. Observations on each individual i obey

the triangular representation

yit ¼ ai þ kit þ ht þ c0xit þ uyit; ð1Þ

Dxit ¼ vit; ð2Þ

where (1, )c0) is a cointegrating vector between yit and xit that is identicalacross individuals. The composite equilibrium error yit ) c0xit is potentiallycomprised of an individual-specific effect ai, an individual-specific lineartrend kit, and a common time-specific factor ht. The remaining idiosyncraticerror uyit is independent across i but possibly dependent across t. An alternativerepresentation for equation (2) allows xit to have an individual-specific vector

of drift terms and for the trend in equation (1) to be induced by this drift. With



some minor modifications, the results of this paper continue to hold in this

alternative environment.4

In addition to individual-specific fixed effects and linear trends, potentially

disparate short-run dynamics of the covariance stationary error process fwyitg ¼

fðuyit; v0itÞ0g introduces an additional source of heterogeneous behaviour across

individuals. The underlying error dynamics are given in Assumption 1.

Assumption 1. (Error Dynamics.) fwyitg is independent across i ¼ 1,…, N,

and has the moving average representation

wyit ¼ Wy

i ðLÞ�yit; ð3Þ

where f�yitg i:i:d: with Eð�yitÞ ¼ 0; Eð�yit�y0it Þ ¼ I, Ejj�yitjj

4 < 1, Wyi ðLÞ ¼P1

j¼0 WyijLj is a (k + 1) · (k + 1) dimensional matrix lag polynomial in the

lag operator L, whereP1j¼0 jjw

ymnij j < 1; and wymn

ij is the m,n-th element of

the matrix Wyij.

Our assumption that wyit is independent across individuals ðE½w

yitw

yjt k� ¼ 0;

i 6¼ j; 1 � k � 1) follows the recent econometrics literature on

non-stationary panel data (e.g. Phillips and Moon, 1999a,b; Kao and Chiang,

2000; Pedroni, 1997). Unlike these authors, we assume that the coefficients in

the Wyi ðLÞ polynomial are fixed for a given i although they can differ across

individuals.

Let Wi be a k + 1-dimensional standard Brownian motion. By Assumption

1, it follows that for each i ¼ 1,…,N, fwyitg obeys the functional central limit

theorem

1ffiffiffiffiT

pX½Tr�t¼1wyit!D

Wyi ð1ÞW i � B

yi ; ð4Þ

as T fi 1, where Byi ¼ ðByui;B0viÞ0is a scaled mixed Brownian motion and

Xyi ¼E½B

yi ð1ÞB

yi ð1Þ

0�¼ Xyuu;i Xy0

vu;i

Xyvu;i Xvv;i

" #¼Wy

i ð1ÞWy0i ð1Þ¼Cy

0;iþX1j¼1

ðCyj;iþCy0

j;iÞ;

Cyj;i ¼ Eðw

yitw

y0it jÞ ¼ E

uyituyit j uyitv

0it j

vituyit vitv

0it j

" #¼ Cy

uu;j;i Cy0vu;j;i

Cyvu;j;i Cvv;j;i

" #:

4The trend in equation (1) can be induced by a drift in {yit}. If instead, there is a drift in {xit} butnone in {yit}, where Dxit ¼ ai + vit with xi0 given, then repeated substitution gives xit ¼ xi0 + ait +nitwhere nit ¼

Ptj¼1 vit is a driftless vector I(1) process. The cointegrating regression becomes

yit ¼ c00xi0 þ c0

1ait þ c0

2nit þ u

yit . In this case, the ensuing analysis is to be performed using the

statistical properties of nit. In computations, one can follow the recommendations of Phillipsand Moon (2000) to obtain an estimate of nit first by estimating the drift aiT ¼1T

PTt¼1 Dxit ¼ 1

T ðxiT xi1Þ and then using nnit ¼ xit xi0 taiT .

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The issues involved in panel cointegration vector estimation and testing

parallels that in the single equation environment. For a single equation, OLS

is a consistent estimator of the cointegrating vector but its asymptotic

distribution depends on the long-run covariance between uyit and vit. Thisnuisance parameter dependency invalidates standard hypothesis testing in the

OLS framework without modifications. DOLS, dynamic generalized least

square (GLS), and fully modified OLS are examples of such modifications.

Similarly, in panel data, Phillips and Moon (1999) and Pedroni (1997) show

that for fixed N, the pooled OLS estimator is a consistent estimator of the

cointegrating vector as T fi 1 and can be used in a first pass in getting

point estimates. In panel data, however, the problems of second-order

asymptotic bias and nuisance parameter dependence are compounded and

are potentially more serious in the sense that the bias accumulates with the

size of the cross-section. In particular, if cOLSNT

is the OLS estimator for the

pooled cross-section time-series data, one cannot rule out the possibility thatffiffiffiffiN

pT ðcOLS

NT cÞ diverges as T fi 1 and then N fi 1. It follows that the

distribution for a Wald statistic for testing linear restrictions becomes even

less useful as the cross-sectional dimension of the panel grows as it too can

diverge.

III. Panel DOLS

We consider the panel DOLS estimator of the vector of slope coefficients cand its limit distribution for various subcases of the model in equations (1) and

(2). We take the model with individual-specific effects as our starting point.5

Section (i) discusses the baseline fixed-effects model. Once we obtain the limit

distribution for this baseline case, the limit theory for more general versions of

the model with heterogeneous linear trends and common time effects follow in

an analogous manner. In section (ii) we add heterogeneous trends to the fixed-

effects model, and section (iii) considers the model with fixed-effects, trends,

and common-time effects.

(i) Fixed effects

In applied work, the researcher will almost always need to include

individual-specific constants in the regression. To handle this situation, we

5Kao and Chiang (2000) derive the sequential limit distribution (T fi 1, N fi 1) for panelDOLS in a model with individual-specific effects. They do not consider the fixed the T fi 1 limittheory with fixed N, nor do they allow for time trends or time-specific effects.



begin by setting ki ¼ 0, ht ¼ 0 for all i and t in equation (1), which we

write as

yit ¼ ai þ c0xit þ uyit: ð5Þ

Assume that uyit is correlated with at most pi leads and lags of vit ¼ Dxit. Tocontrol for this endogeneity, project uyit onto these pi leads and lags

uyit ¼Xpis¼ pi

d0i;svit s þ uit ¼Xpis¼ pi

d0i;sDxit s þ uit ¼ d0izit þ uit; ð6Þ

where di,s is a k · 1 vector of projection coefficients, di ¼ ðd0i; pi ; . . . ;d0i;0; . . . ; d

0i;piÞ

0is a (2pi + 1)k-dimensional vector and zit ¼ ðDx0it pi ; . . . ;

Dx0it; . . . ;Dx0itþpiÞ

0is a (2pi + 1)k-dimensional vector of leads and lags of the

first differences of the variables xit. The projection error uit is, by construction,

orthogonal to all leads and lags of vit. It follows from Assumption 1 that

because wyit is independent across i we project u

yit only onto leads and lags

of Dxit for individual i and not onto leads and lags of the other individuals(Dxjt, j „ i).

Substituting the projection representation for uit in equation (6) into

equation (5) yields

yit ¼ ai þ c0xit þ d0izit þ uit: ð7Þ

The projection defines the new covariance stationary vector process, wit ¼ðuit; v0itÞ

0where for each i,

wit ¼ WiðLÞ�it; WiðLÞ ¼Wuu;iðLÞ 00

0 Wvv;iðLÞ

� �;

and wit obeys the functional central limit theorem

1ffiffiffiffiT

pX½Tr�t¼1wit!

DBi ¼ Wið1ÞW i;

where Bi ¼ ðBui;B0viÞ0, Bui and Bvi are independent, and

Xi ¼ E½Bið1ÞBið1Þ0� ¼Wuu;ið1Þ2 00

0 Wvv;ið1ÞWvv;ið1Þ0� �

¼ Xuu;i 00

0 Xvv;i

� �:

Taking the time-series average of equation (7) gives

1

T

XTt¼1yit ¼ ai þ c0

1

T

XTt¼1xit þ d0i

1

T

XTt¼1zit þ

1

T

XTt¼1uit: ð8Þ

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Subtracting equation (8) from equation (7) eliminates ai and gives

~yyit ¼ c0~xxit þ d0i~zzit þ ~uuit; ð9Þ

where a ‘tilde’ denotes the deviation of an observation from its time-series

average,

~yyit ¼ yit 1

T

XTt¼1yit;

~xxit ¼ xit 1

T

XTt¼1xit;

~zzit ¼ zit 1

T

XTt¼1zit;

~uuit ¼ uit 1

T

XTt¼1uit:

To set up the estimation problem, let ~qqitbe the 2kð1 þ

PNi¼1 piÞ

dimensional vector whose first k elements are ~xxit, elements kð1þPi 1j¼1ð2pj þ 1ÞÞ þ 1 to kð1 þ

Pij¼1ð2pj þ 1ÞÞ are ~zzit and 0s elsewhere.

That is,

~qq1t

¼ ð~xx01t ~zz01t 00 . . . 00Þ0

~qq2t

¼ ð~xx02t 00 ~zz02t . . . 00Þ0

..

. ...

~qqNt

¼ ð~xx0Nt 00 00 . . . ~zz0NtÞ0

:

Let the grand coefficient vector be b ¼ ðc0; d01; . . . ; d0N Þ0and the compact

form of the regression be ~yyit ¼ b0~qqitþ ~uuit. The panel DOLS estimator for the

fixed-effects model is bNT, where

ðbNT

bÞ ¼XNi¼1

XTt¼1

~qqit~qq0it

" # 1 XNi¼1

XTt¼1

~qqit~uuit

" #: ð10Þ

We exploit the fact that the limiting behaviour of the regression error ~uuit, isidentical to that of uit. Some algebra reveals that

1

T

XTt¼1

~xxit~uuit ¼1

T

XTt¼1

~xxituit!D ffiffiffiffiffiffiffiffiffi

Xuu;ip Z

~BBvidWui where ~BBvi ¼ Bvi ZBvi:



We also have

1ffiffiffiffiT

pXTt¼1

~uuit ¼1ffiffiffiffiT

pXTt¼1uit ð1=T Þ 1ffiffiffiffi

Tp

XTt¼1uit!

DBuið1Þ ¼

ffiffiffiffiffiffiffiffiffiXuu;i

pWuið1Þ;

so we are able to use estimated values of ~uuit to obtain a consistent estimate ofXuu,i. The T fi 1 limit theory with fixed N for panel DOLS with individual-

specific fixed effects is given in

Proposition 1. (Fixed N, T fi 1 with fixed effects.) Let ~BBvi ¼ Bvi RBvi.

For the panel DOLS estimator (10), for fixed N as T fi 1,

a. T(cNT ) c) andffiffiffiffiT

pðddi diÞ are independent for each i.

b.ffiffiffiffiN

pT ðc

NT cÞ!D M 1

N mN , where MN ¼ 1N

PNi¼1

R~BBvi~BB

0vi, and mN ¼

1N

PNi¼1


p R~BBvidWui

� �.

c. ½ffiffiffiffiN

pTRðc

NT cÞ�0½RDNR0� 1½

ffiffiffiffiN

pTRðc

NT cÞ�!D v2ðsÞ, where R is an

s · k restriction matrix, DN ¼ M 1N VNM 1

N ; MN ¼ 1N

PNi¼1

R~BBvi~BB

0vi; and

VN ¼ 1N

PNi¼1 Xuu;i

R~BBvi~BB

0vi:

d. DDNT DN !p

0, where DDNT ¼ M 1NT VVNTM

1NT , MNT ¼

1N

PNi¼1

1T 2PTt¼1 ~xxit~xx

0it

�h i; VVNT ¼ 1

N

PNi¼1 XXuu;i 1

T 2PTt¼1 ~xxit~xx

0it

�, and XXuu;i

is a consistent estimator of Xuu,i.

The asymptotic independence of T(cNT ) c) andffiffiffiffiT

pðddi diÞ as T fi 1

follows for the same reasons as in the single-equation environment and

because T(cNT ) c) converges in distribution to a mixed normal randomvector, the limiting chi-square distribution of the quadratic form in part (c) of

the proposition also follows by the standard argument. The asymptotic

covariance matrix DN can be consistently estimated by DNT and it follows that

under the null hypotheses Rc ¼ r, the Wald statistic

½ffiffiffiffiN

pT ðRc

NT rÞ�0½RDDNTR

0� 1½ffiffiffiffiN

pT ðRc

NT rÞ�!D v2ðsÞ ð11Þ

as T fi 1 for any given N.

The sequential limit distribution of cNT is obtained by showing thatthe sequence f

R~BBvi~BB

0vig

1i¼1 obeys a law of large numbers for independent

but heterogeneously distributed observations and that the sequence

fffiffiffiffiffiffiffiffiXuui

p R~BBvi dWuig1i¼1 obeys a central limit theorem for independent but

heterogeneously distributed observations. The sequential limit theory for panel

DOLS is given in Proposition 2.

Proposition 2. (Sequential limit distribution, fixed effects.) For the panelDOLS estimator (10), as T fi 1 and then N fi 1,

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a. C 1=2N

ffiffiffiffiN

pT ðc

NT cÞA Nð0; IkÞ; where CN ¼ ðC 1=2

N ÞðC 1=2N Þ0 ¼

M 1N VNM

1N ; MN ¼ 1

6N

PNi¼1 Xvv;i; and VN ¼ 1

6N

PNi¼1 Xuu;iXvv;i:

b. DDNT CN !p

0, where DDNT is defined in Proposition 1(d).

Controlling for fixed effects results in a shrinkage of the sequential limit

asymptotic variance, compared to when there are no fixed effects. In the case

without fixed effects where ai ¼ 0 for all i, MN ¼ 12N

PNi¼1Xvv;i and

VN ¼ 12N

PNi¼1 Xuu;iXvv;i.

(ii) Fixed effects and heterogeneous trends

We now admit both individual-specific fixed effects and heterogeneous time

trends into the specification. Upon substitution of the projection representation

for the equilibrium error (6) into (1) (with ht ¼ 0 for all t) we have,

yit ¼ ai þ kit þ c0xit þ dizit þ uit: ð12Þ

The time-series average of equation (12) yields

1

T

XTt¼1yit ¼ ai þ ki

T þ 12

� �þ c0

1

T

XTt¼1xit þ d0i

1

T

XTt¼1zit þ

1

T

XTt¼1uit; ð13Þ

where we use the fact that 1TPTt¼1 t ¼ ðT þ 1Þ=2. To control for the fixed-

effects subtract equation (13) from equation (12) to get

~yyit ¼ ki~tt þ c0~xxit þ d0i~zzit þ ~uuit; ð14Þ

where again we use a ‘tilde’ to denote the deviation of an observation from its

time-series average,

~yyit ¼ yit 1

T

XTt¼1yit;

~xxit ¼ xit 1

T

XTt¼1xit;

~zzit ¼ zit 1

T

XTt¼1zit;

~uuit ¼ uit 1

T

XTt¼1uit;

~tt ¼ t T þ 12

:



To set up panel DOLS, let kN ¼ (k1, k2,…, kN)¢ be the vector of trend-slope coefficients, b ¼ ðc0; k0N ; d01; . . . ; d0N Þ

0be the grand coefficient

vector, and define

~qq01t

¼ ð~xx01t ~tt 0 � � � 0 ~zz01t 00 � � � 00Þ0

~qq02t

¼ ð~xx02t 0 ~tt � � � 0 00 ~zz02t � � � 00Þ0

..

. ...

~qq0Nt

¼ ð~xx0Nt 0 0 � � � ~tt 00 00 � � � ~zz0NtÞ0

: ð15Þ

Then the panel DOLS estimator of b is:

bNT

¼XNi¼1

XTt¼1

~qqit~qq0it

" # 1 XNi¼1

XTt¼1

~qqit~yyit

" #: ð16Þ

For fixed N, as T fi 1,ffiffiffiffiT

pðddi diÞ is independent of T(cNT ) c) and

T 3=2ðkkN kN Þ for the standard reasons but T(cNT ) c) and T 3=2ðkkN kN Þremain correlated. The T fi 1 limit theory with fixed N for the fixed-effects

model with trend is given in Proposition 3.

Proposition 3. (Fixed N, T fi 1, fixed effects and trends.) Let ~BBvi ¼Bvi

RBvi. For the panel DOLS estimator (16), for fixed N as T fi 1,

a.ffiffiffiffiT

pðddi diÞ is independent of T(cNT ) c) and T 3=2ðkkN kN Þ for each i.

b.T ðc

NT cÞ

T 3=2ðkkN kN Þ

� �!D M 1

N mN , where MN ¼ M11;N M021;N

M21;N M22;N

� �;

M11;N ¼ 1N

PNi¼1

R~BBvi~BB

0vi;M22;N ¼ 1

12IN ;

M021;N ¼ 1ffiffiffi

Np

Rr~BBv1 1

2

R~BBv1

� �; . . . ; 1ffiffiffi

Np

Rr~BBvN 1

2

R~BBvN

� �h i, and

mN¼" 1ffiffiffi

Np

PNi¼1


p R~BBvidWuih ffiffiffiffiffiffiffiffiffiffi

Xuu;1p R

rdWu1 Wu1ð1Þ2

h i;...;

ffiffiffiffiffiffiffiffiffiffiffiXuu;N

p RrdWuN WuN ð1Þ2

h ii0#:

When T fi 1 then N fi 1, the panel DOLS estimator of the trend-slopecoefficients and the cointegration vector are independent which results in

considerable simplification. The sequential limit theory for panel DOLS in this

case is given in Proposition 4.

Proposition 4. (Sequential limits, fixed effects and trends.) For the panelDOLS estimator (16), as T fi 1 then N fi 1,

a.ffiffiffiffiN

pT ðc

NT cÞ and T 3=2ðkkN kN Þ are independent.

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b. C 1=2N

ffiffiffiffiN

pT ðc

NT cÞA Nð0; IkÞ, where CN ¼ ðC1=2

N ÞðC1=2N Þ0 ¼ M

111;N �

V11;NM 111;N , M11;N ¼ 1

6N

PNi¼1 Xvv;i; and V11;n ¼ 1

6N

PNi¼1 Xuu;iXvv;i:

c. DDNT CN !p

0, where DDNT ¼ M 1NT VVNTM

1NT , MNT ¼ 1

NT 2PNi¼1

PTt¼1 ~xxit~xx

0it; VVNT ¼ 1

NT 2PNi¼1

PTt¼1 XXuui~xxit~xx

0it; and XXuu;i is a consistent

estimator of Xuu,i

Notice that the sequential limit distribution forffiffiffiffiN

pT ðc

NT cÞ is

identical to that obtained in Proposition 2 in the absence of trends.

Construction of a Wald test under the sequential limit theory can proceed as

in section (i).

(iii) Fixed effects, heterogeneous trends, and common time effects

The asymptotic distribution theory that we employ requires that observations

are independent across individuals but in applications, one typically

encounters some degree of CSD. In this section we take up the complete

model (1) which allows us to model a limited form of CSD in which the

equilibrium error for each individual is driven in part by ht.Begin by substituting the projection representation for uyit into equation (1)

to get

yit ¼ ai þ kit þ ht þ c0xit þ d0izit þ uit: ð17Þ

Controlling for the common time effect requires an analysis of the cross-

sectional average of the observations. Because we admit heterogeneity in the

projection coefficients di across i, the resulting cross-sectional averages willinvolve sums such as

PNj¼1 d0jzjt which complicates estimation of the dj

coefficients. The estimation problem can be simplified by proceeding

sequentially and addressing the endogeneity correction separately from

cointegration vector estimation.

To do this, let yzit be the error from projecting each element of yit onto

nit ¼ ð1; t; z0itÞ0and xzit ¼ xit Uinit be the vector of projection errors from

projecting each element of xit onto nit, where Ui is a (k + 2) · pi matrix of

projection coefficients. Substituting the projection representations for yit and

xit into equation (17) gives

yzit ¼ c0xzit þ ht þ uit: ð18Þ

We now work with equation (18) as for the purposes of estimating and

drawing inference about c it is equivalent to equation (17). Now take the

cross-sectional average of equation (18) to get



1

N

XNj¼1yzjt ¼ c0

1

N

XNj¼1xzjt

" #þ ht þ

1

N

XNj¼1ujt: ð19Þ

Subtracting equation (19) from equation (18) eliminates the common time

effect giving

yz�it ¼ c0xz�it þ u�it; ð20Þ

where an ‘asterisk’ denotes the deviation of an observation from its cross-

sectional average. That is,

yz�it ¼ yzit 1NPNj¼1yzjt;

xz�it ¼ xzit 1NPNj¼1xzjt;

u�it ¼ uit 1NPNj¼1ujt:

The panel DOLS estimator of c is

cNT

¼XNi¼1

XTt¼1xz�it x

z�0it

" # 1 XNi¼1

XTt¼1xz�it y

z�it

" #: ð21Þ

As in the case of the fixed-effects model with linear trends, the panel

DOLS estimator of the grand coefficient vector converges to a mixed normal

random vector but T(cNT ) c) and T 3=2ðkkN kN Þ are asymptotically

correlated for fixed N as T fi 1. We omit a statement of the limit theoryfor this case. As T fi 1 then N fi 1, however, T(cNT ) c) and

T 3=2ðkkN kN Þ are independent and the limit theory for this case is givenin Proposition 5.

Proposition 5. (Sequential limit distribution.) For the panel DOLS estimator(21), as T fi 1 then N fi 1,

a.ffiffiffiffiN

pT ðc

NT cÞ and T 3=2ðkkN kN Þ are independent.

b. C 1=2N

ffiffiffiffiN

pT ðc

NT cÞA Nð0; IKÞ, where CN ¼ ðC1=2

N ÞðC1=2N Þ0 ¼

M 111;NV11;NM

111;N ; M11;N ¼ 1

6N

PNi¼1 Xvv;i; and V11;N ¼ 1

6N

PNi¼1Xuu;iXvv;i:

c. DDNT CN !p

0 where DDNT ¼ M 111;NT VV11;NTM

111;NT , M11;NT ¼

1N

PNi¼1

1T 2PTt¼1 x

z�it x

z�0it

h i, VV11;NT ¼ 1

N

PNi¼1

ffiffiffiffiffiffiffiffiffiXXuu;i

q1T 2PTt¼1 x

z�it x

z�0it

h i; and

XXuu;i is a consistent estimator of Xuu,i.

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Notice that the limit distribution of Proposition 5 is identical to the

sequential limit distribution of Proposition 4. Controlling for fixed effects

again produces a shrinkage of the asymptotic variance, while controlling for

the common time effect requires taking the deviation from the cross-sectional

average. These cross-sectional transformations have no effect on the

sequential asymptotic variance of the estimator.

If the modifications to OLS are successful in removing the correlation

between the equilibrium error uyit and the leads and lags of Dxjt for j ¼ 1,…,Nbut the time-specific effects do not fully account for CSD, then the residual

cross-sectional correlation in the projection error uit changes only the formula

for the asymptotic standard errors.6 This is a feasible estimation strategy for a

small to moderate N. But if there remains a correlation between the

equilibrium error and the leads and lags of other equation Dxjt, i „ j, then

panel DOLS exhibits the same sort of second-order asymptotic bias as

pooled OLS as discussed in section II. For small to moderate N, a feasible

solution to this problem is to include leads and lags of Dxjt, j ¼ 1,…,N in theprojection (6).

We close this section by noting that for large N, modelling CSD in panel

data is itself an active area of research and one that has shown itself to be a

thorny problem. What one seeks in this case is a simple parametric structure

that does an adequate job of capturing the long-run covariance structure. Bai

and Ng (2001), Moon and Perron (2002), and Phillips and Sul (2002) study

models in which the error terms in dynamic panel data regressions have a

factor structure, but the implications for such factor models have not been

studied in the panel cointegration context.

IV. Monte-Carlo experiments

In this section, we present some Monte-Carlo experiments to investigate some

small sample properties of panel DOLS and to compare them with single-

equation DOLS in the presence of individual fixed effects and CSD.7 Our data

generating process (DGP) includes two regressors in the cointegrating relation

and is given by

6In this case, the asymptotic variance of panel DOLS is consistently estimated by�PTt¼1 XtX

0t

� 1�PTt¼1 XtXuu;TX

0��PTt¼1 XtX

0t

� 1where Xt ¼ (x1t,…, xNt) and Xuu,T is a consistent

estimator of the long-run covariance matrix of uit, i ¼ 1,…,N.7Kao and Chiang (2000) compared the small-sample performance of panel DOLS and panel fully

modified OLS with fixed effects in the case of a single regressor. They found that panel dynamic OLSperformed much better than panel fully modified OLS in removing finite sample bias so we do notinclude panel fully modified OLS in the comparison.



yit ¼ ai þ c1x1;it þ c2x2;it þ git;

Dx1;it ¼ ai þ v1;it;Dx2;it ¼ v2;it:

Letting wit ¼ (git, v1,it, v2,it)0, �it ¼ (�1,it, �2,it, �3,it)0, and eit ¼ (e1,it, e2,it, e3,it)

0,the short-run dynamics are given by

wit ¼ Aiwit 1 þ �it;

�it ¼ffiffiffiffi/

pht þ

ffiffiffiffiffiffiffiffiffiffiffiffi1 /

peit;

where for j ¼ 1, 2, 3, i ¼ 1,…,N, ej;it i:i:d:Nð0;r2jiÞ; ht ¼ ðh1t; h2t; h3tÞ0,hjt i:i:d:Nð0;r2hjÞ.We designed the DGP to provide a connection to the empirical work on

money demand of the next section, where the regressors are real income

(which has a drift), and the nominal interest rate (which does not).

Accordingly, we induce a trend into the cointegrating relation through the

drift term ai for the first regressor x1,it and specify the second regressor x2,it to

be a driftless I(1) process. In addition, we model the equilibrium error ai + git,to admit a more general form of CSD than the common time effect model

considered in the previous section.8 This single-factor model of the short-run

innovations is of the type considered by Phillips and Sul (2002). CSD in the

equilibrium errors is induced by h1t, while h2t and h3t induce cross-sectionalendogeneity between xj,kt and u

yit, j „ i, k ¼ 1, 2. These features were not

explicitly accounted for in the theoretical analysis, but may be encountered in

empirical work. In the presence of heterogeneous CSD, subtracting off the

cross-sectional average does not completely eliminate CSD. Our interest here

is in evaluating the seriousness of the resulting distortions. The degree of CSD

is modulated by the size of /.The true value of the cointegration vector is (c1, c2) ¼ (1.0, 0.1). For each

individual i, the values of ai, Ai, and rji are first obtained by a draw from

the uniform distribution then held fixed throughout the experiment. The

persistence in the short-run dynamics are controlled by varying the support of

the uniform distribution from which the elements of Ai are drawn. We

consider three levels of persistence and three alternative degrees of CSD.

Persistence levels can be low (A11,i U[0.3,0.5]), medium (A11,i U[0.5,0.7]), or

8In the common time effect specification, the cross-sectional correlation between individuals iand j is identical for all i, j. This homogeneous CSD is obtained here by setting A11,i to beidentical across i. That allowing for heterogeneity in A11,i results in heterogeneous CSD can beseen in the case of an AR(1) where A11,i ¼ qi and all other elements of Ai are set to 0. Then

it can be shown that Corrðgit; gjtÞ ¼ cij ¼ beij

ffiffiffiffiffiffiffiffiffi1 q2i

p ffiffiffiffiffiffiffiffiffi1 q2j

p1 qiqj

, where bij ¼E �1;it�1;jtð Þ

E �21;itð ÞE �2

1;jtð Þ� �1

2

¼/r2h1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

/r2h1 þð1 /Þr2ið Þ /r2h1 þð1 /Þr2jð Þp .

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high (A11,i U[0.7,0.9]), and degrees of CSD are either none (/ ¼ 0), low

(/ ¼ 0.3) or high (/ ¼ 0.7). Assignment of the remaining parameter values are

determined by, A21,i U[)0.05,0.05], A12,i U[)0.05,0.05], A23,i U[)0.05,0.05],

A22,i U[0.0,0.4], A33,i U[0.0,0.04], ai U[23,53] · 10)3, r21i U½1;33� � 10 3,

r22i U½0:25;1:34� � 10 3, r23i U½2:3;57�, and r2hj ¼ ð1=NÞPNi¼1 r2ji. The

long-run variance Xuu,i is estimated by the prewhitened quadratic spectral

(QSPW) method suggested by Sul, Phillips and Choi (2002). Each experiment

consists of 5,000 random samples of T ¼ 40, 100, or 200 observations on

N ¼ 10 or 20 individuals. The number of leads and lags of Dxit included are2 (T ¼ 40), 3 (T ¼ 100), and 4 (T ¼ 200). We organize the experiments

according to the following three cases.

Case 1. (No CSD, Variable Persistence). Setting / ¼ 0 yields no CSD.Persistence levels are low, medium and high.

Case 2. (Homogeneous CSD and Persistence). Setting A11,i ¼ A11,j yields thehomogeneous CSD as in the common time effect specification. We consider highand low CSD and low, medium and high levels of persistence.

Case 3. (Heterogeneous CSD and Persistence). Allowing A11,i „ A11,j yieldsheterogeneous CSD. We consider high and low CSD and low, medium and highlevels of persistence.

We begin with the effective size of nominal 5% and 10% sized tests of the

hypothesis H0: c1 ¼ 1 and H0: c2 ¼ 0.1. To provide a point of comparison,

Table 1 displays the effective size of (single-equation) DOLS tests. Table 2

presents the panel DOLS size results for Case 1. Under low and medium

levels of persistence, the tests are reasonably sized. Size accuracy is seen to

TABLE 1

Effective size of DOLS tests

T Persistence

H0: c1 ¼ 1 H0: c2 ¼ 0.1

5% 10% 5% 10%

40 Low 0.097 0.152 0.093 0.148

Medium 0.118 0.179 0.114 0.174

High 0.179 0.241 0.175 0.239

100 Low 0.100 0.155 0.093 0.151

Medium 0.110 0.170 0.104 0.163

High 0.184 0.250 0.182 0.248

200 Low 0.067 0.128 0.067 0.124

Medium 0.071 0.132 0.074 0.127

High 0.117 0.181 0.114 0.180



improve with increasing sample size both in the time series as well as in the

cross-sectional dimensions. Under high levels of persistence, the test for c1remains reasonably sized but the test for c2 becomes slightly mis-sized. Thismis-sizing worsens somewhat as the cross-section increases (e.g. for T ¼ 100,

the 5% test has size of 16% for N ¼ 10 and 25% for N ¼ 20). In comparison

with DOLS, the test for c1 is better sized whereas the test for c2 is roughlyequivalent.

Table 3 reports the effective size of panel DOLS tests under Case 2. For a

low degree of CSD, the size of the test for c1 improves with persistence and isaccurate when the level of persistence is high.9 The length of the time series is

relatively unimportant. Similar results are obtained for the test on c2 under lowCSD. Under high CSD, there is some mis-sizing of the test on c1, which iscomparable with the size of the DOLS test. For the test on c2, size accuracyimproves with the size of the cross-section and overall size distortion is

modest.

Effective size performance of panel DOLS tests under Case 3, shown in

Table 4, is very similar to that under Case 2. Subtracting off the cross-sectional

average works reasonably well as a control for the heterogeneous CSD

considered here.

Table 5 reports quantiles of cc1 from DOLS and panel DOLS under Case 3.Here, it is seen that dramatic precision gains over single-equation DOLS can

be attained in small samples. For T ¼ 40, N ¼ 10 under high persistence and

high CSD, the inter-95 percentile range for DOLS is ()0.304; 2.495) while for

TABLE 2

Effective size of panel DOLS tests. Case 1: No CSD, variable persistence

T Persistence

H0: c1 ¼ 1 H0: c2 ¼ 0.1

N ¼ 10 N ¼ 20 N ¼ 10 N ¼ 20

5% 10% 5% 10% 5% 10% 5% 10%

40 Low 0.092 0.154 0.087 0.142 0.096 0.156 0.082 0.142

Medium 0.076 0.133 0.065 0.126 0.103 0.169 0.104 0.167

High 0.056 0.098 0.039 0.074 0.110 0.180 0.155 0.237

100 Low 0.072 0.122 0.059 0.109 0.071 0.126 0.064 0.118

Medium 0.062 0.111 0.054 0.106 0.080 0.136 0.072 0.127

High 0.051 0.093 0.042 0.091 0.113 0.184 0.159 0.250

200 Low 0.060 0.105 0.060 0.110 0.058 0.113 0.064 0.115

Medium 0.059 0.102 0.058 0.108 0.062 0.121 0.067 0.117

High 0.045 0.097 0.044 0.085 0.090 0.167 0.147 0.228

9This is largely a feature of Sul, Phillips and Choi’s (2002) QSPW estimator of the long-runvariance which works well under high persistence.

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panel DOLS is (0.883; 1.152). Precision gains continue to accrue when

T ¼ 200. For N ¼ 10, under high persistence and high CSD, the panel DOLS

inter-95 percentile range of (0.979; 1.028) whereas for DOLS it is (0.89;

1.122). Precision advantages are also seen to accrue from enlarging the cross-

sectional dimension. Under high persistence and high CSD, T ¼ 40, the inter-

95 percentile range shrinks from (0.883; 1.152) for N ¼ 10 to (0.924; 1.102)

for N ¼ 20.

Table 6 displays analogous quantile information for cc2. Here, the benefitsfrom the cross-section dimension are largely obtained with N ¼ 10. For

T ¼ 40, under high persistence and CSD, the inter-95 percentile range of

panel DOLS is (0.096; 0.110), which is an improvement over the (0.054;

0.170) range for DOLS.

We summarize the Monte-Carlo results with four general observations.

First, for given T, the empirical size of the panel DOLS t-tests worsens slightly

when N is increased from 10 to 20. Secondly, size distortion, while not

particularly severe at T ¼ 40 is reasonably small at T ¼ 200. Thirdly,

subtracting the cross-sectional average to control for CSD works reasonably

TABLE 3

Effective size of panel DOLS tests. Case 2: homogeneous CSD

T CSD Persistence

H0: c1 ¼ 1 H0: c2 ¼ 0.1

N ¼ 10 N ¼ 20 N ¼ 10 N ¼ 20

5% 10% 5% 10% 5% 10% 5% 10%

40 Low Low 0.103 0.160 0.111 0.182 0.106 0.165 0.091 0.157

Medium 0.087 0.147 0.088 0.150 0.127 0.201 0.083 0.142

High 0.069 0.113 0.051 0.092 0.165 0.245 0.069 0.117

100 Low Low 0.067 0.128 0.079 0.136 0.073 0.129 0.065 0.117

Medium 0.063 0.118 0.073 0.124 0.084 0.143 0.059 0.113

High 0.050 0.094 0.053 0.098 0.140 0.227 0.067 0.129

200 Low Low 0.068 0.131 0.063 0.115 0.074 0.128 0.050 0.098

Medium 0.068 0.122 0.057 0.110 0.075 0.127 0.045 0.090

High 0.047 0.098 0.046 0.097 0.120 0.204 0.053 0.103

40 High Low 0.105 0.167 0.161 0.234 0.168 0.240 0.199 0.267

Medium 0.096 0.152 0.131 0.210 0.167 0.246 0.151 0.217

High 0.065 0.112 0.082 0.139 0.155 0.230 0.088 0.135

100 High Low 0.094 0.155 0.133 0.206 0.100 0.161 0.124 0.195

Medium 0.087 0.147 0.123 0.186 0.113 0.179 0.113 0.178

High 0.064 0.115 0.107 0.173 0.150 0.241 0.098 0.161

200 High Low 0.102 0.164 0.130 0.198 0.089 0.145 0.091 0.146

Medium 0.105 0.163 0.128 0.200 0.090 0.152 0.083 0.144

High 0.087 0.147 0.124 0.191 0.134 0.216 0.080 0.144



well even in the presence of heterogenous CSD. Fourthly, panel DOLS is

much more precise than single-equation DOLS.

V. Long-run money demand

We now employ panel DOLS to estimate coefficients of the long-run M1

demand function. Economists have long been interested in obtaining precise

estimates of money demand for at least two reasons. First, knowing the

income elasticity of money demand helps in determining the rate of

monetary expansion that is consistent with long-run price level stability.

Secondly, knowing the interest elasticity of money demand aids in calculating

the area under the demand curve and to assess the welfare costs of long-run

inflation (Baily, 1956). Additionally, because a stable money-demand func-

tion is a building block of the IS-LM model, economists have historically

been interested in knowing how well this particular aspect of the model

performed. While this motive has become less important in the era of

dynamic general equilibrium models, Lucas (1988) shows that such a

TABLE 4

Effective size of panel DOLS tests. Case 3: heterogeneous CSD

T CSD Persistence

H0: c1 ¼ 1 H0: c2 ¼ 0.1

N ¼ 10 N ¼ 20 N ¼ 10 N ¼ 20

5% 10% 5% 10% 5% 10% 5% 10%

40 Low Low 0.104 0.161 0.101 0.166 0.098 0.159 0.091 0.156

Medium 0.083 0.134 0.082 0.140 0.109 0.172 0.096 0.155

High 0.059 0.104 0.050 0.085 0.131 0.198 0.124 0.198

100 Low Low 0.075 0.134 0.082 0.140 0.071 0.128 0.068 0.123

Medium 0.069 0.121 0.076 0.131 0.085 0.144 0.077 0.132

High 0.051 0.099 0.051 0.099 0.112 0.187 0.138 0.229

200 Low Low 0.042 0.092 0.061 0.113 0.040 0.088 0.055 0.105

Medium 0.040 0.082 0.057 0.112 0.046 0.088 0.056 0.104

High 0.040 0.076 0.042 0.091 0.066 0.131 0.110 0.190

40 High Low 0.114 0.172 0.147 0.217 0.152 0.221 0.206 0.283

Medium 0.094 0.142 0.118 0.190 0.147 0.215 0.167 0.234

High 0.058 0.098 0.053 0.094 0.142 0.213 0.089 0.147

100 High Low 0.098 0.165 0.138 0.202 0.098 0.165 0.137 0.207

Medium 0.088 0.141 0.131 0.199 0.108 0.173 0.131 0.191

High 0.061 0.110 0.082 0.138 0.146 0.225 0.095 0.159

200 High Low 0.090 0.150 0.128 0.199 0.068 0.119 0.102 0.164

Medium 0.088 0.147 0.145 0.207 0.074 0.131 0.102 0.166

High 0.070 0.124 0.119 0.190 0.108 0.188 0.087 0.145

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neoclassical model with a cash in advance constraint generates a standard

money demand function.

We follow Stock and Watson (1993), Ball (2001), and Hoffman et al.

(1995) and approach long-run money demand as a cointegrating relationship.

Our analysis suggests that instability exhibited by time-series estimates from

the literature do not reflect underlying shifts in behavioural relationships

but instead indicate inherent difficulties associated with estimation using

TABLE 5

Quantiles for first regressor slope (Case 3): panel DOLS and DOLS

CSD Persistence

DOLS (N ¼ 10) Panel DOLS (N ¼ 10) Panel DOLS (N ¼ 20)

2.5% 50% 97.5% 2.5% 50% 97.5% 2.5% 50% 97.5%

T ¼ 40

None Low 0.460 0.999 1.512 0.952 1.001 1.048 0.976 1.000 1.025

Med 0.225 0.999 1.769 0.934 1.001 1.070 0.967 1.002 1.038

High )0.192 1.010 2.420 0.890 1.007 1.137 0.937 1.008 1.078

Low Low 0.456 0.999 1.529 0.951 1.000 1.052 0.973 1.000 1.029

Med 0.206 0.996 1.782 0.931 1.001 1.075 0.963 1.003 1.044

High )0.259 1.006 2.435 0.882 1.010 1.150 0.936 1.011 1.090

High Low 0.432 0.999 1.536 0.956 1.000 1.047 0.971 1.000 1.031

Med 0.181 0.996 1.783 0.936 1.001 1.073 0.960 1.003 1.049

High )0.304 1.004 2.495 0.883 1.011 1.152 0.924 1.012 1.102

T ¼ 100

None Low 0.906 1.000 1.094 0.988 1.000 1.012 0.994 1.000 1.006

Med 0.852 1.000 1.146 0.982 1.000 1.018 0.991 1.000 1.010

High 0.692 1.002 1.347 0.966 1.003 1.039 0.981 1.002 1.022

Low Low 0.904 1.000 1.095 0.985 1.000 1.016 0.991 1.000 1.009

Med 0.850 1.000 1.150 0.978 1.000 1.023 0.988 1.001 1.014

High 0.682 1.001 1.369 0.961 1.005 1.052 0.978 1.004 1.030

High Low 0.901 1.000 1.099 0.986 1.000 1.014 0.991 1.000 1.009

Med 0.847 0.999 1.154 0.978 1.000 1.022 0.987 1.001 1.015

High 0.666 1.000 1.378 0.957 1.005 1.056 0.974 1.004 1.036

T ¼ 200

None Low 0.970 1.000 1.029 0.996 1.000 1.004 0.998 1.000 1.002

Med 0.955 1.000 1.045 0.994 1.000 1.006 0.997 1.000 1.004

High 0.898 1.000 1.107 0.987 1.001 1.014 0.993 1.000 1.008

Low Low 0.970 1.000 1.030 0.993 1.000 1.007 0.996 1.000 1.004

Med 0.955 1.000 1.047 0.990 1.000 1.010 0.995 1.000 1.006

High 0.895 1.000 1.117 0.983 1.002 1.024 0.991 1.001 1.012

High Low 0.969 1.000 1.031 0.994 1.000 1.006 0.996 1.000 1.004

Med 0.952 1.000 1.048 0.990 1.000 1.010 0.994 1.000 1.007

High 0.890 1.001 1.122 0.979 1.002 1.028 0.987 1.002 1.018



relatively short sample spans in environments with persistent short-run

dynamics. Combining observations across countries allows us to obtain

relatively sharp and stable estimates of money demand elasticities and the

panel cointegration approach seems well suited to take up King’s (1988)

suggestion to extend the money-demand analysis beyond the US. In his

words, ‘the results of such investigations would provide us with sharper

estimates of the long run values of Friedman’s (1956) ‘‘‘numerical constants

TABLE 6

Quantiles for second regressor slope (Case 3): panel DOLS and DOLS

CSD Persistence

DOLS (N ¼ 10) Panel DOLS (N ¼ 10) Panel DOLS (N ¼ 20)

2.5% 50% 97.5% 2.5% 50% 97.5% 2.5% 50% 97.5%

T ¼ 40

None Low 0.072 0.100 0.129 0.098 0.100 0.102 0.099 0.100 0.101

Med 0.065 0.101 0.142 0.098 0.100 0.103 0.098 0.100 0.101

High 0.050 0.105 0.172 0.097 0.102 0.107 0.094 0.098 0.101

Low Low 0.075 0.100 0.127 0.098 0.100 0.102 0.099 0.100 0.101

Med 0.068 0.101 0.140 0.098 0.101 0.103 0.098 0.100 0.102

High 0.052 0.105 0.172 0.097 0.102 0.108 0.094 0.098 0.102

High Low 0.076 0.100 0.126 0.098 0.100 0.102 0.098 0.100 0.102

Med 0.069 0.101 0.139 0.097 0.101 0.104 0.097 0.100 0.103

High 0.054 0.105 0.170 0.096 0.103 0.110 0.092 0.098 0.104

T ¼ 100

None Low 0.093 0.100 0.107 0.099 0.100 0.101 0.100 0.100 0.100

Med 0.091 0.100 0.111 0.099 0.100 0.101 0.099 0.100 0.100

High 0.085 0.102 0.128 0.099 0.101 0.102 0.098 0.099 0.101

Low Low 0.094 0.100 0.106 0.099 0.100 0.101 0.100 0.100 0.100

Med 0.092 0.100 0.110 0.099 0.100 0.101 0.099 0.100 0.101

High 0.086 0.102 0.125 0.099 0.101 0.103 0.098 0.099 0.101

High Low 0.094 0.100 0.106 0.099 0.100 0.101 0.099 0.100 0.101

Med 0.092 0.100 0.110 0.099 0.100 0.101 0.099 0.100 0.101

High 0.087 0.102 0.124 0.099 0.101 0.104 0.097 0.099 0.101

T ¼ 200

None Low 0.097 0.100 0.103 0.100 0.100 0.100 0.100 0.100 0.100

Med 0.096 0.100 0.105 0.100 0.100 0.100 0.100 0.100 0.100

High 0.094 0.101 0.113 0.100 0.100 0.101 0.099 0.100 0.100

Low Low 0.102 0.100 0.098 0.100 0.100 0.100 0.100 0.100 0.100

Med 0.103 0.100 0.096 0.100 0.100 0.100 0.100 0.100 0.100

High 0.105 0.099 0.089 0.100 0.100 0.099 0.101 0.100 0.100

High Low 0.102 0.100 0.098 0.100 0.100 0.100 0.100 0.100 0.100

Med 0.103 0.100 0.096 0.100 0.100 0.099 0.100 0.100 0.100

High 0.105 0.099 0.090 0.101 0.099 0.098 0.101 0.100 0.099

676 Bulletin


of monetary behavior’’’ when we approach the difficult problem of the short

run demand for money’’.

The equation that we estimate is,

lnMitPit

� �¼ ai þ kit þ ht þ cy ln Yit þ crRit þ u

yit ð22Þ

for i ¼ 1,… , 19, where Mit is an M1 measure of money, Pit is the price level,

Yit is real GDP, and Rit is a nominal short-term interest rate. Data definitions

and sources are available in the unpublished appendix. In addition to country

specific effects, ai, we allow for possibly heterogeneous linear trends and

common time effects. These trends are included to capture changes in the

financial technology that affects money demand independent of income and

the opportunity cost of holding money.

(i) Pre-testing: cointegration and homogeneity restrictions

Panel DOLS estimation of equation (22) requires that the equilibrium errors

are stationary and that the cointegrating vectors for each country must be

identical. To investigate the stationarity of the equilibrium errors, we employ

Pedroni’s (1999) panel-t test. This results in the rejection of the null

hypothesis of no cointegration at the 0.1% level whether heterogeneous linear

trends and common time effects are included or not.10

Next, we conduct a Wald test of the homogeneity restrictions on the

cointegrating vector. When trends are omitted from the regression, the evi-

dence against homogeneity is mixed. The asymptotic test rejects the restric-

tions in this case, but in some unreported Monte-Carlo experiments, we found

moderate size distortion in the Wald test for sample sizes of N ¼ 19 and

T ¼ 40. Using a size adjustment from these experiments, the homogeneity

restrictions on income is rejected at the 5% level but not for the interest rate

(P ¼ 0.30). However, when we impose homogeneity on the interest rate

slope, the test for slope homogeneity on income is not rejected (P ¼ 0.70).

When heterogeneous linear trends are included, the evidence supporting

homogeneity strengthens. Here, we obtain a P-value of 0.63 for the test of

homogeneity on the income coefficient and a P-value of 0.22 for the test

of homogeneity on the interest rate coefficient.

10We also confirmed these cointegration test results by using Im, Pesaran and Shin (1997) andMaddala and Wu (1999) panel unit root tests under the assumption that the cointegrating vector isknown to be (1, )1.0,0.05). The justification for using these values is that 1.0 is a typical value of theincome elasticity estimated in the literature while a common estimate of the interest rate semielasticity )0.05.



(ii) Comparison between single-equation and panel DOLS estimates

Our panel DOLS estimates use two leads and two lags of D ln Yit and DRit inthe regressions. Point estimates and asymptotic standard errors are reported in

Table 7.

Single-equation DOLS estimates are seen to display such cross-sectional

variability that they are difficult to interpret. In DOLS regressions without

trend, the income elasticities are all positive, ranging from 0.134 (Belgium) to

a whopping 2.64 (Norway), but the interest semi-elasticity has the wrong sign

for Belgium, France, Ireland, and Japan. When a trend is included in the

regression, income elasticity estimates are negative for Finland, Iceland,

Norway, and New Zealand, and interest semi-elasticity estimates are positive

for Finland, France, and Iceland. If we maintain an underlying belief that

the financial systems and transactions technologies across modern economies

are essentially similar, the cross-sectional variability in these estimates must

reflect the inherent difficulty of obtaining good estimates rather than evidence

of disparate economic behavior.

TABLE 7

Single-equation and panel dynamic OLS estimates of long-run money demand

Country

No trend With trend

ccy (s.e.) ccR (s.e.) ccy (s.e.) ccR (s.e.) Trend

Austria 0.901 (0.139) )0.009 (0.029) 1.552 (0.349) )0.037 (0.026) )0.001Belgium 0.134 (0.218) 0.009 (0.039) 1.183 (0.444) )0.033 (0.026) 0.002

Denmark 1.460 (0.170) )0.043 (0.009) 0.684 (0.321) )0.036 (0.006) 0.009

Finland 1.019 (0.634) )0.006 (0.114) )0.740 (0.881) 0.009 (0.011) )0.005France 0.677 (0.213) 0.010 (0.020) 0.842 (0.699) 0.004 (0.031) 0.003

Germany 1.548 (0.033) )0.019 (0.008) 1.691 (0.197) )0.023 (0.009) 0.003

Iceland 0.594 (0.161) )0.010 (0.005) )0.451 (1.093) )0.004 (0.008) 0.004

Ireland 0.507 (0.169) 0.022 (0.022) 1.670 (2.805) 0.015 (0.029) 0.005

Netherlands 1.112 (0.111) )0.045 (0.020) 0.309 (0.415) )0.011 (0.022) 0.003

Norway 2.641 (0.450) )0.160 (0.046) )0.676 (2.154) )0.092 (0.060) 0.013

Portugal 0.517 (0.136) )0.037 (0.017) 1.624 (0.379) )0.043 (0.011) 0.010

Spain 1.203 (0.091) )0.030 (0.008) 1.203 (0.190) )0.030 (0.009) 0.003

Switzerland 1.020 (0.208) )0.062 (0.021) 1.447 (0.482) )0.053 (0.021) 0.011

UK 1.738 (0.097) )0.089 (0.008) 2.128 (0.726) )0.089 (0.008) 0.016

Japan 0.889 (0.599) 0.009 (0.200) 1.798 (0.415) )0.076 (0.061) 0.016

Australia 0.926 (0.136) )0.043 (0.012) 0.068 (0.329) )0.048 (0.007) 0.002

New Zealand 1.349 (0.539) )0.076 (0.026) )1.233 (1.149) )0.084 (0.018) )0.001Canada 1.245 (0.219) )0.057 (0.024) 2.420 (0.903) )0.078 (0.024) )0.013US 0.428 (0.074) )0.035 (0.008) 1.022 (0.417) )0.039 (0.007) )0.001

Panel 0.860 (0.092) )0.020 (0.007) 1.079 (0.264) )0.022 (0.006) —

Panela 0.820 (0.105) )0.017 (0.005) 0.986 (0.336) )0.016 (0.005) —

Note: aControls for common time effect.

678 Bulletin


Panel DOLS estimates are shown at the bottom of Table 7. When the panel

regression omits trends, we estimate 0.86 (asymptotic s.e. ¼ 0.09) and the

interest semi-elasticity to be )0.02 (asymptotic s.e. ¼ 0.01). When we include

heterogeneous trends, we estimate the income elasticity to be 1.08 (asymptotic

s.e. ¼ 0.26) and the interest semi-elasticity to be )0.02 (asymptotic

s.e. ¼ 0.01). Results obtained from controlling for CSD are very similar.

To further illustrate the problem of estimation instability in the time

dimension, we constructed recursive single-equation DOLS coefficient

estimates for the US, UK, France, and Japan and panel DOLS for all 19

countries. Recursive DOLS estimates from 1979 to 1995 for both the income

elasticity and interest semi-elasticity exhibit substantially more variability than

the recursive panel DOLS estimates and in several instances even change sign.

These results are also contained in the unpublished appendix.

VI. Conclusions

Heterogeneity and persistence in short-run dynamics can create substantial

variability in single-equation cointegration vector point estimates. The result is

that these estimators can be quite sensitive to the particular time span of the

observations as well as to the particular individual being studied. This small

sample fragility can be encountered in spite of the superconsistency of these

estimators.

In these environments, panel DOLS can provide much more precise

estimates. Panel DOLS is straightforward to compute and relevant test

statistics have standard asymptotic distributions. The asymptotic distributions

were found to provide reasonably close approximations to the exact sampling

distributions in small samples.

We applied the panel DOLS method to estimate the long-run money

demand function using a panel of 19 countries with annual data from 1957 to

1996. The estimates in which we have the most confidence are an income

elasticity near 1 and an interest rate semi-elasticity of )0.02.

Final Manuscript received: April 2003

References

Baba, Y., Hendry, D. F. and Starr, R. M. (1992). ‘The demand for M1 in the U.S.A., 1960–

1988’, Review of Economic Studies, Vol. 59, pp. 25–61.

Bai, J. and Ng, S. (2001). ‘Determining the number of factors in approximate factor models’,

Econometrica, Vol. 70, pp. 191–221.

Bailey, M. J. (1956). ‘The welfare cost of inflationary finance’, Journal of Political Economy,

Vol. 64, pp. 93–110.



Ball, L. (2001). ‘Another look at long-run money demand’, Journal of Monetary Economics,

Vol. 47, pp. 37–44.

Friedman, M. (1956). ‘The quantity theory of money – a restatement’, in Friedman M. (ed.),

Studies in the Quantity Theory of Money, University of Chicago Press, Chicago.

Gandolfi, A. E. and Lothian, J. R. (1976). ‘The demand for money from the great depression to

the present’, American Economic Review, Vol. 66, pp. 46–51.

Hoffman, D. L., Rasche, R. H. and Tieslau, M. A. (1995). ‘The stability of long-run money

demand in five industrial countries’, Journal of Monetary Economics, Vol. 35, pp. 317–339.

Im, K. S., Pesaran, M. H. and Shin, Y. (1997). Testing for Unit Roots in Heterogeneous Panels,

Discussion Paper, University of Cambridge, Cambridge.

Inder, B. (1993). ‘Estimating long-run relationships in economics: a comparison of different

approaches’, Journal of Econometrics, Vol. 57, pp. 53–68.

Kao, C. and Chiang, M.-H. (2000). ‘On the estimation and inference of a cointegrated

regression in panel data’, in Baltagi B. H. (ed.), Advances in Econometrics: Nonstationary

Panels, Panel Cointegration and Dynamic Panels, Vol. 15, pp. 179–222.

King, R. G. (1988). Money Demand in the United States: A Quantitative Review: A Comment,

Carnegie-Rochester Conference Series on Public Policy, Vol. 29, pp. 169–172.

Lucas, R. E. Jr (1988). Money Demand in the United States: A Quantitative Review, Carnegie-

Rochester Conference Series on Public Policy, Vol. 29, pp. 137–168.

Maddala, G. S. and Wu, S. (1999). ‘A comparative study of unit root tests with panel data and a

new simple test’, Oxford Bulletin of Economics and Statistics, Vol. 61, pp. 631–652.

Meltzer, A. H. (1963). ‘The demand for money: a cross-section study of business firms’,

Quarterly Journal of Economics, Vol. 77, pp. 405–422.

Moon, H. R. and Perron, P. (2002). ‘Testing for a unit root in panels with dynamic factors’,

University of Southern California, mimeo.

Mulligan, C. B. (1997). ‘Scale economies, the value of time, and the demand for money:

longitudinal evidence from firms’, Journal of Political Economy, Vol. 105, pp. 1061–1079.

Mulligan, C. B. and Sala-i-Martin, X. (1992). ‘U.S. money demand: surprising cross-sectional

estimates’, Brookings Papers on Economic Activity, No. 2, pp. 285–329.

Pedroni, P. (1997). ‘Fully modified OLS for heterogeneous cointegrated panels and the case of

purchasing power parity’, Department of Economics, Indiana University, mimeo.

Pedroni, P. (1999). ‘Critical values for cointegration tests in heterogeneous panels with multiple

regressors’, Oxford Bulletin of Economics and Statistics, Vol. 61, pp. 653–670.

Phillips, P. C. B. and Moon, H. R. (1999). ‘Linear regression limit theory for nonstationary

panel data’, Econometrica, Vol. 67, pp. 1057–1111.

Phillips, P. C. B. and Moon, H. R. (2000). ‘Nonstationary panel data analysis: an overview of

some recent development’, Econometric Reviews, Vol. 19, pp. 263–286.

Phillips, P. C. B. and Sul, D. (2002). ‘Dynamic panel estimation and homogeneity testing under

cross section dependence’, Yale University, mimeo.

Saikkonen, P. (1991). ‘Asymptotically efficient estimation of cointegration regressions’,

Econometric Theory, Vol. 7, pp. 1–21.

Stock, J. H. and Watson, M. W. (1993). ‘A simple estimator of cointegrating vectors in higher

order integrated systems’, Econometrica, Vol. 61, pp. 783–820.

Sul, D., Phillips, P. C. B. and Choi, C.-Y. (2002). ‘Prewhitening Bias in HAC Estimation’, Yale

University, mimeo.

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